# Introduction to Groups and Geometries

## Section2.4Group homomorphisms

### Subsection2.4.1

#### Definition2.4.1.Group homomorphism.

Let $$G,H$$ be groups.
A map $$\phi\colon G\to H$$ is called a homomorphism if
\begin{equation*} \phi(xy) = \phi(x)\phi(y) \end{equation*}
for all $$x,y$$ in $$G\text{.}$$ A homomorphism that is both injective (one-to-one) and surjective (onto) is called an isomorphism of groups. If $$\phi\colon G\to H$$ is an isomorphism, we say that $$G$$ is isomorphic to $$H\text{,}$$ and we write $$G\approx H\text{.}$$
Show that each of the following are homomorphisms.
• $$GL(n,\R)\to \R^\ast$$ given by $$M\to \det M$$
• $$\Z\to \Z$$ given by $$x\to mx\text{,}$$ some fixed $$m\in \Z$$
• $$G\to G\text{,}$$ $$G$$ any group, given by $$x\to axa^{-1}\text{,}$$ some fixed $$a\in G$$
• $$\C^\ast\to\C^\ast$$ given by $$z\to z^2$$
Show that each of the following are not homomorphisms. In each case, demonstrate what fails.
• $$\Z\to \Z$$ given by $$x\to x+3$$
• $$\Z\to \Z$$ given by $$x\to x^2$$
• $$D_4\to D_4$$ given by $$g\to g^2$$

#### Definition2.4.3.Kernel of a group homomorphism.

Let $$\phi\colon G\to H$$ be a group homomorphism, and let $$e_H$$ be the identity element for $$H\text{.}$$ We write $$\ker(\phi)$$ to denote the set
\begin{equation*} \ker(\phi) :=\phi^{-1}(e_H) = \{g\in G\colon \phi(g)=e_H\}, \end{equation*}
called the kernel of $$\phi\text{.}$$
Find the kernel of each of the following homomorphisms.
• $$\C^\ast\to \C^\ast$$ given by $$z\to z^n$$
• $$\Z_8\to \Z_8$$ given by $$x\to 6x \pmod{8}$$
• $$G\to G\text{,}$$ $$G$$ any group, given by $$x\to axa^{-1}\text{,}$$ some fixed $$a\in G$$
1. $$\displaystyle C_n$$
2. $$\displaystyle \langle 4\rangle = \{0,4\}$$
3. $$\displaystyle \{e\}$$

#### Definition2.4.8.Normal subgroup.

A subgroup $$H$$ of a group $$G$$ is called normal if $$ghg^{-1}\in H$$ for every $$g\in G\text{,}$$ $$h\in H\text{.}$$ We write $$H\trianglelefteq G$$ to indicate that $$H$$ is a normal subgroup of $$G\text{.}$$

### Exercises2.4.2Exercises

#### Basic properties of homomorphisms.

##### 1.
Prove Properties 1 and 2.
##### 2.
Prove Properties 3 and 4.
##### 3.
Prove Properties 5, 6, and 7.
Hint.
Use [cross-reference to target(s) "constructequivrelpart" missing or not unique].

#### 4.

Show that the inverse of an isomorphism is an isomorphism.

#### Proof of the First Isomorphism Theorem.

##### 5.
Hint.
First, suppose $$K=\ker(\phi)$$ for some homomorphism $$\phi\colon G \to G'\text{.}$$ Explain why Item 6 of Proposition 2.4.5 can be rephrased to say that there is a one-to-one correspondence $$G/K \leftrightarrow \phi(G)$$ given by $$gK\leftrightarrow \phi(g)\text{.}$$ Now use the bijection $$G/K\leftrightarrow \phi(G)$$ to impose the group structure of $$\phi(G)$$ (Item 4 of Proposition 2.4.5) on $$G/K\text{.}$$ Conversely, if $$G/K$$ is a group with the group operation (2.4.1), define $$\phi\colon G\to G/K$$ by $$\phi(g)=gK\text{,}$$ then check that $$\phi$$ is a homomorphism and that $$\ker(\phi)=K\text{.}$$

#### 7.

Let $$n,a$$ be relatively prime positive integers. Show that the map $$\Z_n\to \Z_n$$ given by $$x\to ax$$ is an isomorphism.
Hint.
Use the fact that $$\gcd(m,n)$$ is the least positive integer of the form $$sm+tn$$ over all integers $$s,t$$ (see Exercise 2.3.2.4). Use this to solve $$ax=1 \pmod{n}$$ when $$a,n$$ are relatively prime.

#### 8.Another construction of $$\Z_n$$.

Let $$n\geq 1$$ be an integer and let $$\omega=e^{i2\pi/n}\text{.}$$ Let $$\phi\colon \Z\to S^1$$ be given by $$k\to \omega^k\text{.}$$
1. Show that the the image of $$\phi$$ is the group $$C_n$$ of $$n$$th roots of unity.
2. Show that $$\phi$$ is a homomorphism, and that the kernel of $$\phi$$ is the set $$n\Z=\{nk\colon k\in \Z\}\text{.}$$
3. Conclude that $$\Z/\!(n\Z)$$ is isomorphic to the group of $$n$$-th roots of unity.

#### 9.Isomorphic images of generators are generators.

Let $$S$$ be a subset of a group $$G\text{.}$$ Let $$\phi\colon G\to H$$ be an isomorphism of groups, and let $$\phi(S)=\{\phi(s)\colon s\in S\}\text{.}$$ Show that $$\phi(\langle S\rangle)=\langle \phi(S)\rangle\text{.}$$

#### 10.Conjugation.

Let $$G$$ be a group, let $$a$$ be an element of $$G\text{,}$$ and let $$C_a\colon G\to G$$ be given by $$C_a(g)=aga^{-1}\text{.}$$ The map $$C_a$$ is called conjugation by the element $$a$$ and the elements $$g,aga^{-1}$$ are said to be conjugate to one another.
1. Show that $$C_a$$ is an isomorphism of $$G$$ with itself.
2. Show that "is conjugate to" is an equivalence relation. That is, consider the relation on $$G$$ given by $$x\sim y$$ if $$y=C_a(x)$$ for some $$a\text{.}$$ Show that this is an equivalence relation.

#### 11.Isomorphism induces an equivalence relation.

Prove that "is isomorphic to" is an equivalence relation on groups. That is, consider the relation $$\approx$$ on the set of all groups, given by $$G\approx H$$ if there exists a group isomorphism $$\phi\colon G\to H\text{.}$$ Show that this is an equivalence relation.

#### Characterization of normal subgroups.

Prove Proposition 2.4.9. That Item 1 is equivalent to Item 2 is established by Proposition 2.4.6.
##### 13.
Show that Item 3 implies Item 2. The messy part of this proof is to show that multiplication of cosets is well-defined. This means you start by supposing that $$xK=x'K$$ and $$yK=y'K\text{,}$$ then show that $$xyK=x'y'K\text{.}$$
##### 14.Further characterizations of normal subgroups.
Show that Item 3 is equivalent to the following conditions.
1. $$gKg^{-1}= K$$ for all $$g\in G$$
2. $$gK = Kg$$ for all $$g\in G$$

#### 15.Automorphisms.

Let $$G$$ be a group. An automorphism of $$G$$ is an isomorphism from $$G$$ to itself. The set of all automorphisms of $$G$$ is denoted $$\Aut(G)$$.
1. Show that $$\Aut(G)$$ is a group under the operation of function composition.
2. Show that
\begin{equation*} \Inn(G) := \{C_g\colon g\in G\} \end{equation*}
is a subgroup of $$\Aut(G)\text{.}$$ (The group $$\Inn(G)$$ is called the group of inner automorphisms of $$G\text{.}$$)
3. Find an example of an automorphism of a group that is not an inner automorphism.